Question

Find the centroid for an area defined by the equations y = x² + 3 and y = - (x – 2)² + 7

Answer 1

The centroid for a bounded region is defined as:


`x_c = \frac{ \int\limits^a_b {x(f(x) - g(x))} dx}{ \int\limits^a_b {(f(x) -g(x))} \, dx} \\ \\ y_c = \frac{ \int\limits^a_b {(1)/(2)(f(x) +g(x))(f(x) - g(x))} dx}{ \int\limits^a_b {(f(x) -g(x))} \, dx}`
where

`f(x) = -(x-2)^2+7 = -x^2+4x+3 \\ \\ g(x) = x^2 +3`
limits are where f(x) = g(x) at x = 0,2
Subbing into the integrals

`x_c = (\int_0^2 (-2x^3+4x^2) dx)/(\int_0^2 (-2x^2+4x) dx) = 1 \\ \\ y_c = (\int_0^2 (-4x^3+2x^2+12x) dx)/(\int_0^2 (-2x^2+4x) dx) = 5`

The centroid is the point (1,5)

Answer 2

The points where the 2 graphs intersect is where x = 0 and x = 2.

- 2 2
x = INT x dA / INT dA
0 0

INT dA = INT -x^2 + 4x + 3 - (x^2 + 3 ) dx = INT -2x^2 + 4x
= -2 x^3/3 + 2x^2
= 2.667 between 0 and 2

xdA = -2x^3 + 4x^2 INT xdA = -x^4/2 + 4x^3/3 = 2.667

centroid = 2.667 / 2.667 = 1 (x = 1)